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Chapter 10: Problem 46

Find the indefinite integral. $$ \int\left(\ln t \mathbf{i}+\frac{1}{t} \mathbf{j}+\mathbf{k}\right) d t $$

Short Answer

Expert verified
The integral of the given vector field is \((t \ln t - t + C) \mathbf{i} + (\ln | t | + C') \mathbf{j} + (t + C'')\mathbf{k}\).

Step by step solution

01

Deconstruct the Vector Field

Separate the vector field into its components: \(\ln t\), \(\frac{1}{t}\) and 1, and treat them as separate integrals: \(\int \ln t \, dt\), \(\int \frac{1}{t} \, dt\), and \(\int dt\) respectively.
02

Integrate Each Component

To find the indefinite integral of the first component, use the integral of the natural logarithm, which is \(t \ln t - t + C\). The integral of \(\frac{1}{t}\) is \(\ln | t | + C'\) and the integral of a constant 1 with respect to \(t\) is simply \(t + C''\). Remember to include the constant of integration \(C\), \(C'\), and \(C''\) in each integral.
03

Combine the Components

Combine the results of the three integrals back into one vector. This gives the resulting vector field: \((t \ln t - t + C) \mathbf{i} + (\ln | t | + C') \mathbf{j} + (t + C'')\mathbf{k}\).

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Most popular questions from this chapter

Prove the property. In each case, assume that \(\mathbf{r}, \mathbf{u},\) and \(\mathbf{v}\) are differentiable vector-valued functions of \(t,\) \(f\) is a differentiable real-valued function of \(t,\) and \(c\) is a scalar. If \(\mathbf{r}(t) \cdot \mathbf{r}(t)\) is a constant, then \(\mathbf{r}(t) \cdot \mathbf{r}^{\prime}(t)=0\)

Find the indefinite integral. $$ \int\left[(2 t-1) \mathbf{i}+4 t^{3} \mathbf{j}+3 \sqrt{t} \mathbf{k}\right] d t $$

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The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+\sqrt{9-t^{2}} \mathbf{k} $$

Use the definition of the derivative to find \(\mathbf{r}^{\prime}(t)\). $$ \mathbf{r}(t)=\langle 0, \sin t, 4 t\rangle $$
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